Rotation operator propagators for time-varying radiofrequency pulses in NMR spectroscopy: applications to shaped pulses and pulse trains.

The propagator for trains of radiofrequency pulses can be directly integrated numerically or approximated by average Hamiltonian approaches. The former provides high accuracy and the latter, in favorable cases, convenient analytical formula. The Euler-angle rotation operator factorization of the propagator provides insights into performance that are not as easily discerned from either of these conventional techniques. This approach is useful in determining whether a shaped pulse can be represented over some bandwidth by a sequence τ1-Rϕ(β)-τ2, in which Rϕ(β) is a rotation by an angle β around an axis with phase ϕ in the transverse plane and τ1 and τ2 are time delays, allowing phase evolution during the pulse to be compensated by adjusting time periods prior or subsequent to the pulse. Perturbation theory establishes explicit formulas for τ1 and τ2 as proportional to the average transverse magnetization generated during the shaped pulse. The Euler-angle representation of the propagator also is useful in iterative reduction of pulse-interrupted-free precession schemes. Application to Carr-Purcell-Meiboom-Gill sequences identifies an eight-pulse phase alternating scheme that generates a propagator nearly equal to the identity operator.